Trumans History in MCM

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Trumans History in MCM

• First entered 1989 - 3rd year of contest.
• Has participated every year since.
• 37 teams
• 14 awards (8 honorable mentions, 5 meritorious
  awards, 1 winner)

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• This presentation contains advice that I feel has
  worked for me. I dont suggest that the advice I
  give is the only way to advise a team or even the
  best way to advise a team, but simply reflects my
  personal approach and experience.
• Steven J. Smith Truman State University Math
  Department

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1. Recruiting a Team

• A. Posting signs around math classrooms
  w/contest directions telling them to contact me
  (especially early years of contest but still do
  this to round out team).
• B. Announcements by faculty in classes - my
  classes and others especially early years of
  contest.
• C. Informally collect input of faculty -
  especially from faculty who teach required
  mid/upper level courses such as linear algebra,
  analysis I algebra I followed up by me
  talking to students in the hallways or by me
  e-mailing students
• D. They will find you if you run the contest
  for a couple of years in a row.
• E. Team members of incomplete teams can help
  recruit remaining members

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• F. Contest Description to Students
• Hard (There are hundreds of teams of bright
  students. However we have managed to win 14
  awards, so it isnt impossible.)
• Intense (4 straight days w/ long hours of work
  and writing.)
• Fun (If you like to be challenged. You get to
  work as part of a team.)
• Rewards creativity reasoning (Problems from
  practical settings and are open-ended. You can
  use books, journals, web software as resources.
  Often practical assumptions need to be chosen by
  team.)
• All majors allowed (not just math).
• Luck is involved (perhaps a team member recently
  studied related material on one of the two stated
  contest problems).
• Experience with programming and software is a
  plus
• Frustrating (Hard, intense activities usually
  choose are frustrating. Plus, every once in a
  while a good solution doesnt get chosen for an
  award.)
• Classes come first! (You have to work around your
  classes.)
• Check it out yourself (http//www.comap.com/under
  graduate/contests/mcm/MCM-ICM08.pdf)

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2. An Ideal Team Member

• A. Bright, mathematically creative
  hard-working.
• B. Persevering and not easily intimidated by
  math problems.
• C. Humble.
• D. Works well in a group and isnt domineering.
• E. Has done the contest before (but not a
  necessity).
• F. Does at least one of the following writes
  well, produces good documents, programs well or
  knows software well.

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3. An Ideal Team

• A. Has ideal members with complementary
  strengths.
• B. Has at least one CS major, or has a member
  with the skills of a CS major.
• C. Gets along with each other.
• D. Has at least one person who writes well.
• E. Has at least one person who can produce a high
  quality document.

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4. The First Meeting (Orientation)

• A. Pass out some old contest problems to team and
  pose some questions concerning how to approach
  them.
• i. Does this problem require mathematical
  reasoning and creativity combined with a degree
  of simulation or is it a problem with a known
  optimal technique?

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Mathematical Reasoning Creativity

• 2007 A Gerrymandering
• Gerrymandering The United States Constitution
  provides that the House of Representatives shall
  be composed of some number (currently 435) of
  individuals who are elected from each state in
  proportion to the states population relative to
  that of the country as a whole. While this
  provides a way of determining how many
  representatives each state will have, it says
  nothing about how the district represented by a
  particular representative shall be determined
  geographically. This oversight has led to
  egregious (at least some people think so, usually
  not the incumbent) district shapes that look
  unnatural by some standards.
• Hence the following question Suppose you were
  given the opportunity to draw congressional
  districts for a state. How would you do so as a
  purely baseline exercise to create the
  simplest shapes for all the districts in a
  state? The rules include only that each district
  in the state must contain the same population.
  The definition of simple is up to you but you
  need to make a convincing argument to voters in
  the state that your solution is fair. As an
  application of your method, draw geographically
  simple congressional districts for the state of
  New York.

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Known optimal technique

• 1989 A - The Midge Classification Problem
• Two species of midges, Af and Apf, have been
  identified by biologists Grogan and Wirth on the
  basis of antenna and wing length (see Figure 1).
  It is important to be able to classify a specimen
  as Af of Apf, given the antenna and wing length.
• Given a midge that you know is species Af or Apf,
  how would you go about classifying it?
• Apply your method to three specimens with
  (antenna, wing) lengths (1.24,1.80),(1.28,1.84),(1
  .40,2.04).
• Assume that the species is a valuable pollinator
  and species Apf is a carrier of a debilitating
  disease. Would you modify your classification
  scheme and if so, how?

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1993 A - Optimal Composting An environmentally
conscious institutional cafeteria is recycling
customers' uneaten food into compost by means of
microorganisms. Each day, the cafeteria blends
the leftover food into a slurry, mixes the slurry
with crisp salad wastes from the kitchen and a
small amount of shredded newspaper, and feeds the
resulting mixture to a culture of fungi and soil
bacteria, which digest slurry, greens, and papers
into usable compost. The crisp green provide
pockets of oxygen for the fungi culture, and the
paper absorbs excess humidity. At times, however,
the fungi culture is unable or unwilling to
digest as much of the leftovers as customers
leave the cafeteria does not blame the chef for
the fungi culture's lack of appetite. Also, the
cafeteria has received offers for the purchase of
large quantities of it compost. Therefore, the
cafeteria is investigating ways to increase its
production of compost. Since it cannot yet afford
to build a new composting facility, the cafeteria
seeks methods to accelerate the fungi culture's
activity, for instance, by optimizing the fungi
culture's environment (currently held at about
120 F and 100 humidity), or by optimizing the
composition of the moisture fed to the fungi
culture, or both. Determine whether any relation
exists between the proportions of slurry, greens,
and paper in the mixture fed to the fungi
culture, and the rate at which the fungi culture
composts the mixture. if no relation exists,
state so. otherwise, determine what proportions
would accelerate the fungi culture's activity. In
addition to the technical report following the
format prescribed in the contest instructions,
provide a one-page nontechnical recommendation
for implementation for the cafeteria manager.
Table 1 shows the composition of various mixtures
in pounds of each ingredient kept in separate
bins, and the time that it took the fungi to
culture to compost the mixtures, from the date
fed to the date completely composted table
omitted.
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• ii. Is it essentially a simulation problem?

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Simulation Problem

• 1988 A - The Drug Runner Problem
• Two listening posts 5.43 miles apart pick up a
  brief radio signal. The sensing devices were
  oriented at 110 degrees and 119 degrees,
  respectively, when the signal was detected and
  they are accurate to within 2 degrees. The signal
  came from a region of active drug exchange, and
  it is inferred that there is a powerboat waiting
  for someone to pick up drugs. it is dusk, the
  weather is calm, and there are no currents. A
  small helicopter leaves from Post 1 and is able
  to fly accurately along the 110 degree angle
  direction. The helicopter's speed is three times
  the speed of the boat. The helicopter will be
  heard when it gets within 500 ft of the boat.
  This helicopter has only one detection device, a
  searchlight. At 200 ft, it can just illuminate a
  circular region with a radius of 25 ft.
• Develop an optimal search method for the
  helicopter.
• Use a 95 confidence level in your calculations.

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• iii. Which of the two stated problems look more
  approachable/familiar/friendlier? (
  Although this contradicts the advice of some who
  have written on the contest, personally I feel it
  is better for teams to attempt the more
  approachable/friendlier problem as opposed to the
  problem that looks more challenging/cool. The
  contest is challenging enough by itself.)

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Friendly Problem (to a past team of mine).

• 2006 A - Positioning and Moving Sprinkler Systems
  for Irrigation
• There are a wide variety of techniques available
  for irrigating a field. The technologies range
  from advanced drip systems to periodic flooding.
  One of the systems that is used on smaller
  ranches is the use of "hand move" irrigation
  systems. Lightweight aluminum pipes with
  sprinkler heads are put in place across fields,
  and they are moved by hand at periodic intervals
  to insure that the whole field receives an
  adequate amount of water. This type of irrigation
  system is cheaper and easier to maintain than
  other systems. It is also flexible, allowing for
  use on a wide variety of fields and crops. The
  disadvantage is that it requires a great deal of
  time and effort to move and set up the equipment
  at regular intervals.
• Given that this type of irrigation system is to
  be used, how can it be configured to minimize the
  amount of time required to irrigate a field that
  is 80 meters by 30 meters? For this task you are
  asked to find an algorithm to determine how to
  irrigate the rectangular field that minimizes the
  amount of time required by a rancher to maintain
  the irrigation system. One pipe set is used in
  the field. You should determine the number of
  sprinklers and the spacing between sprinklers,
  and you should find a schedule to move the pipes,
  including where to move them.
• A pipe set consists of a number of pipes that can
  be connected together in a straight line. Each
  pipe has a 10 cm inner diameter with rotating
  spray nozzles that have a 0.6 cm inner diameter.
  When put together the resulting pipe is 20 meters
  long. At the water source, the pressure is 420
  Kilo- Pascals and has a flow rate of 150 liters
  per minute. No part of the field should receive
  more than 0.75 cm per hour of water, and each
  part of the field should receive at least 2
  centimeters of water every 4 days. The total
  amount of water should be applied as uniformly as
  possible.

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Cool Problem (to same past team).

• 2006 B - Wheel Chair Access at Airports
• One of the frustrations with air travel is the
  need to fly through multiple airports, and each
  stop generally requires each traveler to change
  to a different airplane. This can be especially
  difficult for people who are not able to easily
  walk to a different flight's waiting area. One of
  the ways that an airline can make the transition
  easier is to provide a wheel chair and an escort
  to those people who ask for help. It is generally
  known well in advance which passengers require
  help, but it is not uncommon to receive notice
  when a passenger first registers at the airport.
  In rare instances an airline may not receive
  notice from a passenger until just prior to
  landing.
• Airlines are under constant pressure to keep
  their costs down. Wheel chairs wear out and are
  expensive and require maintenance. There is also
  a cost for making the escorts available.
  Moreover, wheel chairs and their escorts must be
  constantly moved around the airport so that they
  are available to people when their flight lands.
  In some large airports the time required to move
  across the airport is nontrivial. The wheel
  chairs must be stored somewhere, but space is
  expensive and severely limited in an airport
  terminal. Also, wheel chairs left in high traffic
  areas represent a liability risk as people try to
  move around them. Finally, one of the biggest
  costs is the cost of holding a plane if someone
  must wait for an escort and becomes late for
  their flight. The latter cost is especially
  troubling because it can affect the airline's
  average flight delay which can lead to fewer
  ticket sales as potential customers may choose to
  avoid an airline.
• Epsilon Airlines has decided to ask a third party
  to help them obtain a detailed analysis of the
  issues and costs of keeping and maintaining wheel
  chairs and escorts available for passengers. The
  airline needs to find a way to schedule the
  movement of wheel chairs throughout each day in a
  cost effective way. They also need to find and
  define the costs for budget planning in both the
  short and long term.
• Epsilon Airlines has asked your consultant group
  to put together a bid to help them solve their
  problem. Your bid should include an overview and
  analysis of the situation to help them decide if
  you fully understand their problem. They require
  a detailed description of an algorithm that you
  would like to implement which can determine where
  the escorts and wheel chairs should be and how
  they should move throughout each day. The goal is
  to keep the total costs as low as possible. Your
  bid is one of many that the airline will
  consider. You must make a strong case as to why
  your solution is the best and show that it will
  be able to handle a wide range of airports under
  a variety of circumstances.
• Your bid should also include examples of how the
  algorithm would work for a large (at least 4
  concourses), a medium (at least two concourses),
  and a small airport (one concourse) under high
  and low traffic loads. You should determine all
  potential costs and balance their respective
  weights. Finally, as populations begin to include
  a higher percentage of older people who have more
  time to travel but may require more aid, your
  report should include projections of potential
  costs and needs in the future with
  recommendations to meet future needs.

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• B. Explain to team that there are always two
  problems One called the continuous problem and
  one called the discrete problem. The continuous
  problem is one in which aspects may look more
  related to a calculus class or a differential
  equations class, while the discrete problem is
  one in which aspects may look more related to a
  discrete math class.

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Continuous Problem

• 1994 A - Concrete Slab Floors
• The U.S. Dept. of Housing and Urban Development
  (HUD) is considering constructing dwellings of
  various sizes, ranging from individual houses to
  large apartment complexes. A principal concern is
  to minimize recurring costs to occupants,
  especially the costs of heating and cooling. The
  region in which the construction is to take place
  is temperate, with a moderate variation in
  temperature throughout the year.
• Through special construction techniques, HUD
  engineers can build dwellings that do not need to
  rely on convection- that is, there is no need to
  rely on opening doors or windows to assist in
  temperature variation. The dwellings will be
  single-story, with concrete slab floors as the
  only foundation. You have been hired as a
  consultant to analyze the temperature variation
  in the concrete slab floor to determine if the
  temperature averaged over the floor surface can
  be maintained within a prescribed comfort zone
  throughout the year. If so, what size/shape of
  slabs will permit this?
• Part 1, Floor Temperature Consider the
  temperature variation in a concrete slab given
  that the ambient temperature varies daily within
  the ranges given Table 1. Assume that the high
  occurs at noon and the low at midnight. Determine
  if slabs can be designed to maintain a
  temperature averaged over the floor surface
  within the prescribed comfort zone considering
  radiation only. Initially, assume that the heat
  transfer into the dwelling is through the exposed
  perimeter of the slab and that the top and bottom
  of the slabs are insulated. Comment on the
  appropriateness and sensitivity of these
  assumptions. If you cannot find a solution that
  satisfies Table 1, can you find designs that
  satisfy a Table 1 that you propose?
• Ambient Temperature Comfort Zone High 85 76 Low
  60 65 Part 2, Building Temperature Analyze the
  practicality of the initial assumptions and
  extend the analysis to temperature variation
  within the single-story dwelling. Can the house
  be kept within the comfort zone?
• Part 3, Cost of Construction Suggest a design
  that considers HUD's objective of reducing or
  eliminating heating and cooling costs,
  considering construction restrictions and costs.

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Discrete Problem

• 1991 B - The Steiner Tree Problem
• The cost for a communication line between two
  stations is proportional to the length of the
  line. The cost for conventional minimal spanning
  trees of a set of stations can often be cut by
  introducing "phantom" stations and then
  constructing a new Steiner tree. This device
  allows costs to be cut by up to 13.4 ( 1-
  sqrt(3/4)). Moreover, a network with n stations
  never requires more than n-2 points to construct
  the cheapest Steiner tree. Two simple cases are
  shown in Figure 1. For local networks, it often
  is necessary to use rectilinear or
  "checker-board" distances, instead of straight
  Euclidean lines. Distances in this metric are
  computed as shown in Figure 2.
• Suppose you wish to design a minimum costs
  spanning tree for a local network with 9
  stations. Their rectangular coordinates are
  a(0,15), b(5,20), c(16,24), d(20,20), e(33,25),
  f(23,11), g(35,7), h(25,0) i(10,3). You are
  restricted to using rectilinear lines. Moreover,
  all "phantom" stations must be located at lattice
  points (i.e., the coordinates must be integers).
  The cost for each line is its length.
• Find a minimal cost tree for the network.
• Suppose each stations has a cost wd(3/2), where
  ddegree of the station. If w1.2, find a minimal
  cost tree.
• Try to generalize this problem

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• C. Explain to team sometimes you get the lucky
  problem (a problem related to a class that at
  least one team member recently completed).

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Lucky Problem (to person who just took PDEs)

• 1990 A - The Brain-Drug Problem
• Researches on brain disorders test the effects of
  the new medical drugs -- for example, dopamine
  against Parkinson's disease -- with intracerebral
  injections. To this end, they must estimate the
  size and the shape of the spatial distribution of
  the drug after the injection, in order to
  estimate accurately the region of the brain that
  the drug has affected. The research data consist
  of the measurements of the amounts of drug in
  each of 50 cylindrical tissue samples (see Figure
  1 and Table 1). Each cylinder has length 0.76 mm
  and diameter 0.66 mm. The centers of the parallel
  cylinders lie on a grid with mesh 1mm X 0.76mm X
  1mm, so that the cylinders touch one another on
  their circular bases but not along their sides,
  as shown in the accompanying figure. The
  injection was made near the center of the
  cylinder with the highest scintillation count.
  Naturally, one expects that there is a drug also
  between the cylinders and outside the region
  covered by the samples.
• Estimate the distribution in the region affected
  by the drug.
• One unit represents a scintillation count, or
  4.753e-13 mole of dopamine. For example, the
  table shows that the middle rear cylinder
  contains 28353 units.
• Table 1. Amounts of drug in each of 50
  cylindrical tissue samples. Rear vertical
  section 164 442 1320 414 188 480 7022 14411 5158
  352 2091 23027 28353 13138 681 789 21260 20921
  11731 727 213 1303 3765 1715 453 Front vertical
  section 163 324 432 243 166 712 4055 6098 1048
  232 2137 15531 19742 4785 335 444 11431 14960
  3182 301 294 2061 1036 258 188

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• D. Explain to team that sometimes two very
  different solutions to the same problem can both
  be correct. (In the problem that follows, two
  separate solutions that won awards were both
  wedges but they pointed in opposite directions
  and worked very differently.)

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Differently Approached Problem

• 2003 MCM ProblemsPROBLEM A The Stunt Person
• An exciting action scene in a movie is going to
  be filmed, and you are the stunt coordinator! A
  stunt person on a motorcycle will jump over an
  elephant and land in a pile of cardboard boxes to
  cushion their fall. You need to protect the stunt
  person, and also use relatively few cardboard
  boxes (lower cost, not seen by camera, etc.).
• Your job is to
• determine what size boxes to use
• determine how many boxes to use
• determine how the boxes will be stacked
• determine if any modifications to the boxes would
  help
• generalize to different combined weights (stunt
  person motorcycle) and different jump heights
• Note that, in "Tomorrow Never Dies", the James
  Bond character on a motorcycle jumps over a
  helicopter.

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• E. Explain to team that sometimes we turn in a
  good solution, but it doesnt win an award. The
  competition is tough and there are many good
  teams.

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Good solution with no award problem

• Radio Channel Assignments
• We seek to model the assignment of radio channels
  to a symmetric network of transmitter locations
  over a large planar area, so as to avoid
  interference. One basic approach is to partition
  the region into regular hexagons in a grid
  (honeycomb-style), as shown in Figure 1, where a
  transmitter is located at the center of each
  hexagon.
• Figure 1
• An interval of the frequency spectrum is to be
  allotted for transmitter frequencies. The
  interval will be divided into regularly spaced
  channels, which we represent by integers 1, 2, 3,
  ... . Each transmitter will be assigned one
  positive integer channel. The same channel can be
  used at many locations, provided that
  interference from nearby transmitters is avoided.
  Our goal is to minimize the width of the interval
  in the frequency spectrum that is needed to
  assign channels subject to some constraints. This
  is achieved with the concept of a span. The span
  is the minimum, over all assignments satisfying
  the constraints, of the largest channel used at
  any location. It is not required that every
  channel smaller than the span be used in an
  assignment that attains the span.
• Let s be the length of a side of one of the
  hexagons. We concentrate on the case that there
  are two levels of interference.
• Requirement A There are several constraints on
  frequency assignments. First, no two transmitters
  within distance 4s of each other can be given the
  same channel. Second, due to spectral spreading,
  transmitters within distance 2s of each other
  must not be given the same or adjacent channels
  Their channels must differ by at least 2. Under
  these constraints, what can we say about the span
  in,

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Requirement B Repeat Requirement A, assuming the
grid in the example spreads arbitrarily far in
all directions. Requirement C Repeat
Requirements A and B, except assume now more
generally that channels for transmitters within
distance 2s differ by at least some given integer
k, while those at distance at most 4s must still
differ by at least one. What can we say about the
span and about efficient strategies for designing
assignments, as a function of k? Requirement D
Consider generalizations of the problem, such as
several levels of interference or irregular
transmitter placements. What other factors may be
important to consider? Requirement E Write an
article (no more than 2 pages) for the local
newspaper explaining your findings.
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• F. Discuss How to Use the Internet.
• i. Lost or unsure on how to start on problem?
  Try typing key words into internet search engine.
  
• ii. Books and journals make better references
  than web pages see if you can use internet to
  find a useful journal article/book. Dont just
  use web pages as source.

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• G. Discuss Role of Software
• i. Mathematica, SPSS, Matlab, SASare all
  potential valuable resources.
• ii. Inform them of location of all computer
  manuals/reference books Ive gathered from
  faculty.

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• H. Explain that their classes are their first
  priority and the contest works around their
  classes and their obligations for class.
  However, in order to do well in the contest,
  their best effort after completing their class
  obligations will be necessary.

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• I. Pass out two old contest problems. Their
  assignment is to meet with their team members
  over the weekend and brain storm approaches to
  the problems. The results of that exercise are
  part of the discussion at the next meeting.

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5. Pre-contest meetings.

• A. Discuss results of brainstorming on the two
  dry-run problems.
• B. Provide them with a letter from me explaining
  to instructors that the contest is time intensive
  and any arrangements they can make postponing
  assignments tests for team members is
  appreciated. They can choose to use the letter
  if they prefer.

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• C. Discuss the importance of the write-up.
• i. The write-up is as important as the actual
  solution.
• ii. The summary page is more important than the
  solution. I suspect papers with a poor summary
  page never have more than the summary page read
  by a judge.

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• iii. The instructions provide a preferred format
  for the paper. Do not stray from that format
• Restatement Clarification of the Problem - state
  in your own words what you are going to
  do.Assumptions with Rationale/Justification -
  emphasize those assumptions that bear on the
  problem. List clearly all variables used in your
  model.
• Model Design and justification for type model
  used/developed.
• Model Testing and Sensitivity Analysis, including
  error analysis, etc.
• Discuss strengths and weakness to your model or
  approach.
• Provide algorithms in words, figures, or flow
  charts (as a step by step algorithmic approach)
  for all computer codes developed.

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t

• Each page of the solution should contain the team
  control number and the page number at the top of
  the page we suggest using a page header on each
  page, for example Team 321 Page 6 of 13. ( I
  now require all my teams to know how to do this
  before contest starts).
• iv. Provide them with a copy of one of our best
  past papers along with a copy of a former contest
  winner to help them design the write-up of their
  solution.
• v. Warn them against padding their paper with
  statements that attempt to boost the mathematical
  prowess by discussing techniques/concepts they
  did not come close to using (avoid trying to snow
  the judges, they know better).
• vi. Inform them that graphs and figures which
  help explain ideas are generally well-received
  and preferred over long, long blocks of texts.
  

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D. Explain Sensitivity, Stability and Error
Analysis in Modeling. Included in my discussion
of these topics are examples taken from former
winning papers or good papers of my own past team
members.
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E. Discuss suggested time frame (admittedly my
stated deadlines are later than some others I
have seen suggested but seem more realistic to
me). i. Problem selection Friday noon at
latest, Thursday midnight preferred. ii. Basic
model developed Saturday early afternoon at
latest (refined model and possibly alternate
model by Sunday early evening). iii. Write-up
begin by Sunday afternoon. Immediately after
basic model developed is preferred, then write as
you go. iv. Summary page Introduction
Monday afternoon, but plan on devoting time and
care on this part. Dont attempt to write these
parts until solution is completed. v. Paper
submitted to advisor One hour before deadline
to be on the safe side.
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• F. Discuss division of labor.
• i. Different duties can be undertaken at the
  same time for team members. For example, one
  member might work on refining the model, while
  another is writing code and the other could be
  writing the document.
• ii. All three members dont have to contribute to
  each basic duty of the project. Perhaps one will
  not write code, or perhaps one will not
  contribute to the writing or producing the actual
  document. Each team member can use their own
  strengths.

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• G. Explain arrangements with Public Safety
  concerning access to buildings and rooms 24 hours
  per day.

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6. Post-Contest Meeting.

• A. Usually about 2 weeks after contest, giving
  them some time to rest up and catch up in
  classes.
• B. Main purpose is to have fun.
• C. Gives them a chance to find out what the other
  team did and to talk about what they worked so
  hard on.
• D. Gives me a chance to critique their work in
  person, thank their contribution and find out
  what they did that worked and what didnt work.

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7. Last Comments

• A. Beyond my current job as a math professor,
  Ive previously worked as an instructor and also
  as a coach. My role as MCM advisor has been more
  like that of a coach, providing advice,
  encouragement and resource in addition to
  information and knowledge.
• B. You can learn a lot by reading the judges
  commentary that is included with the printing of
  the outstanding papers.
• C. I feel any team that has given their best
  effort, learned math and possibly submitted a
  solution has had a successful contest. When our
  teams have gone beyond this level of success, I
  feel two of the largest factors have been how
  bright and hard-working our students have been
  and how well our faculty as a group have
  taught/inspired these students in their math
  courses throughout their entire time at the
  university.

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• D. Be clear and upfront about the nature of the
  contest when recruiting team members, you are
  more likely to end up with a dedicated team that
  will have a rewarding experience.
• E. For some students who have done this
  contest, it is the highlight of their
  college experience. The significance of the MCM
  lies in the influence it has upon the contestants
  and as founder Ben Fusaro has noted,

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• "the confidence that this experience engenders.